A planet has radius R = (1/10)Rₑ and the same mean density as Earth. A well of depth R/5 is dug in it, and a wire of the same length with linear mass density 10⁻³ kg m⁻¹ is lowered into the well without touching its sides. Find the force applied at the top of the wire to hold it. Take Rₑ = 6 × 10⁶ m and Earth's surface g = 10 m s⁻².
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Question

A planet has radius R = (1/10)Rₑ and the same mean density as Earth. A well of depth R/5 is dug in it, and a wire of the same length with linear mass density 10⁻³ kg m⁻¹ is lowered into the well without touching its sides. Find the force applied at the top of the wire to hold it. Take Rₑ = 6 × 10⁶ m and Earth's surface g = 10 m s⁻².

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G1 Physics · Accepted Answer

Step 1: Outside a spherical Earth/planet, g(r)=GM/r²; near the surface gh/g≈(R/(R+h))²≈1-2h/R. Inside a uniform sphere g varies linearly with distance from the centre. Step 2: Use the relation appropriate to the stated height/depth. Step 3: Simplify. Final: B: 10⁸ N.